DP IB Maths: AI HL

Revision Notes

3.8.2 Shortest Distances with Lines

Test Yourself

Shortest Distance Between a Point and a Line

How do I find the shortest distance from a point to a line?

  • The shortest distance from any point to a line will always be the perpendicular distance
    • Given a line l  with equation bold r bold space equals bold space bold a plus straight lambda bold b  and a point P not on l
    • The scalar product of the direction vector, b, and the vector in the direction of the shortest distance will be zero
  • The shortest distance can be found using the following steps:
    • STEP 1: Let the vector equation of the line be r and the point not on the line be P, then the point on the line closest to P will be the point F
      • The point F is sometimes called the foot of the perpendicular
    • STEP 2: Sketch a diagram showing the line l and the points P and F
      • The vector stack F P with rightwards arrow on top will be perpendicular to the line l
    • STEP 3: Use the equation of the line to find the position vector of the point F  in terms of λ
    • STEP 4: Use this to find the displacement vector stack F P with rightwards arrow on top in terms of λ
    • STEP 5: The scalar product of the direction vector of the line l and the displacement vector stack F P with rightwards arrow on top will be zero
      • Form an equation stack F P with rightwards arrow on top times bold b equals 0 and solve to find λ
    • STEP 6: Substitute λ into stack F P with rightwards arrow on top and find the magnitude open vertical bar stack F P with rightwards arrow on top close vertical bar 
      • The shortest distance from the point to the line will be the magnitude of stack F P with rightwards arrow on top
  • Note that the shortest distance between the point and the line is sometimes referred to as the length of the perpendicular

7-3-4-foot-of-the-perpendicular

How do we use the vector product to find the shortest distance from a point to a line?

  • The vector product can be used to find the shortest distance from any point to a line on a 2-dimensional plane
  • Given a point, P, and a line r = a + λb
    • The shortest distance from P to the line will befraction numerator open vertical bar stack A P with rightwards arrow on top cross times b close vertical bar blank over denominator open vertical bar b close vertical bar end fraction
    • Where A is a point on the line
    • This is not given in the formula booklet

Exam Tip

  • Column vectors can be easier and clearer to work with when dealing with scalar products.

Worked example

Point A  has coordinates (1, 2, 0) and the line l has equation bold r equals open parentheses table row 2 row 0 row 6 end table close parentheses plus lambda open parentheses table row 0 row 1 row 2 end table close parentheses

Point B lies on the l such that open square brackets A B close square brackets  is perpendicular to l.

Find the shortest distance from A to the line l.

3-10-5-ib-aa-hl-short-distance-lines-we-1

Shortest Distance Between Two Lines

How do we find the shortest distance between two parallel lines?

  • Two parallel lines will never intersect
  • The shortest distance between two parallel lines will be the perpendicular distance between them
  • Given a line l subscript 1 with equation bold r equals bold a subscript 1 plus lambda bold d subscript 1and a line begin mathsize 16px style l subscript 2 end style with equation bold r equals bold a subscript 2 plus mu bold d subscript 2 then the shortest distance between them can be found using the following steps:
    • STEP 1: Find the vector between bold a subscript 1 and a general coordinate from l subscript 2 in terms of μ  
    • STEP 2: Set the scalar product of the vector found in STEP 1 and the direction vector Error converting from MathML to accessible text.equal to zero
      • Remember the direction vectors bold d subscript 1 and are scalar multiples of each other and so either can be used here
    • STEP 3: Form and solve an equation to find the value of μ
    • STEP 4: Substitute the value of μ  back into the equation for l subscript 2 to find the coordinate on l subscript 2 closest to l subscript 1
    • STEP 5: Find the distance between bold a subscript 1 and the coordinate found in STEP 4
  • Alternatively, the formula fraction numerator open vertical bar stack A B with rightwards arrow on top cross times bold d close vertical bar blank over denominator open vertical bar bold d close vertical bar end fraction can be used
    • Where stack A B with rightwards arrow on top is the vector connecting the two given coordinates and bold a subscript 2  
    • d is the simplified vector in the direction of bold d subscript 1 and bold d subscript 2
    • This is not given in the formula booklet

How do we find the shortest distance from a given point on a line to another line?

  • The shortest distance from any point on a line to another line will be the perpendicular distance from the point to the line
  • If the angle between the two lines is known or can be found then right-angled trigonometry can be used to find the perpendicular distance
    • The formula fraction numerator open vertical bar stack A B with rightwards arrow on top cross times bold d close vertical bar blank over denominator open vertical bar bold d close vertical bar end fraction given above is derived using this method and can be used
  • Alternatively, the equation of the line can be used to find a general coordinate and the steps above can be followed to find the shortest distance

How do we find the shortest distance between two skew lines?

  • Two skew lines are not parallel but will never intersect
  • The shortest distance between two skew lines will be perpendicular to both of the lines
    • This will be at the point where the two lines pass each other with the perpendicular distance where the point of intersection would be
    • The vector product of the two direction vectors can be used to find a vector in the direction of the shortest distance
    • The shortest distance will be a vector parallel to the vector product
  • To find the shortest distance between two skew lines with equations bold r equals bold a subscript 1 plus lambda bold d subscript 1 and bold r equals bold a subscript 2 plus mu bold d subscript 2 ,
    • STEP 1: Find the vector product of the direction vectors bold space bold d subscript 1 and bold space bold d subscript 2
      • bold d bold space equals blank bold d subscript 1 blank cross times blank bold d subscript 2
    • STEP 2: Find the vector in the direction of the line between the two general points on begin mathsize 16px style l subscript 1 end style and l subscript 2  in terms of λ  and μ
      • stack A B with rightwards arrow on top space equals blank bold b blank minus blank bold a blank
    • STEP 3: Set the two vectors parallel to each other
      • bold d bold space equals space k stack A B with rightwards arrow on top
    • STEP 4: Set up and solve a system of linear equations in the three unknowns, k comma blank lambda and mu

Exam Tip

  • Exam questions will often ask for the shortest, or minimum, distance within vector questions
  • If you’re unsure start by sketching a quick diagram
  • Sometimes calculus can be used, however vector methods are usually required

Worked example

A drone travels in a straight line and at a constant speed. It moves from an initial point (-5, 4, -8) in the direction of the vector open parentheses negative 1
space space 2
space space 1 close parentheses. At the same time as the drone begins moving a bird takes off from initial point (6, -4, 3) and moves in a straight line at a constant speed in the direction of the vector open parentheses space space 2
minus 3
space space 4 close parentheses.

Find the minimum distance between the bird and the drone during this movement.

3-10-5-ib-aa-hl-short-distance-lines-we-2

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Amber

Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.